Accurate measurement of neutrino energies is essential to many of the scientific goals of large-volume neutrino telescopes. The fundamental observable in such detectors is the Cherenkov light produced by the transit through a medium of charged particles created in neutrino interactions. The amount of light emitted is proportional to the deposited energy, which is approximately equal to the neutrino energy for νe and νμ charged-current interactions and can be used to set a lower bound on neutrino energies and to measure neutrino spectra statistically in other channels. Here we describe methods and performance of reconstructing charged-particle energies and topologies from the observed Cherenkov light yield, including techniques to measure the energies of uncontained muon tracks, achieving average uncertainties in electromagnetic-equivalent deposited energy of ∼ 15% above 10 TeV.